In right-angled triangle DEF, where angle E = 90° and angle F = 45°, what is the exact value of the product sin F × tan F?

Introduction / Context:
This question is a straightforward test of standard trigonometric values for a special angle, 45 degrees. Right-angled triangles with one acute angle equal to 45 degrees are isosceles, and the trigonometric ratios for 45 degrees are well known. The task is to compute sin 45 degrees and tan 45 degrees and then multiply them together correctly.

Given Data / Assumptions:

• Triangle DEF is right-angled at E, so angle E = 90°.
• Angle F = 45°, so the remaining angle D is also 45°.
• We are asked to find sin F × tan F where F = 45°.
• Standard exact values for trigonometric functions at 45° are used.

Concept / Approach:
For an angle of 45 degrees, consider the standard right-angled isosceles triangle with legs of equal length, say 1, and hypotenuse sqrt(2). From this, sin 45° is opposite over hypotenuse, which is 1 / sqrt(2). Tan 45° is opposite over adjacent, which is 1. After recalling these values, we simply multiply sin 45° by tan 45° and simplify the result. Because tan 45° equals 1, the product is just sin 45° itself.

Step-by-Step Solution:
In a 45°–45°–90° triangle, let each leg be 1 and the hypotenuse be sqrt(2).Compute sin 45°. Sin 45° = opposite / hypotenuse = 1 / sqrt(2).Compute tan 45°. Tan 45° = opposite / adjacent = 1 / 1 = 1.Now find the product sin F × tan F: sin 45° × tan 45° = (1 / sqrt(2)) * 1.This equals 1 / sqrt(2). No further simplification is needed unless rationalisation is required, but the option given uses 1/√2.

Verification / Alternative check:
Use approximate decimal values to check. Sin 45° is about 0.707, tan 45° is exactly 1, so their product is about 0.707. The value 1/√2 is approximately 1 / 1.414 ≈ 0.707, which matches. None of the other options produce this value, confirming the correctness of 1/√2 as the exact form.

Why Other Options Are Wrong:

• √2 is about 1.414, which is the reciprocal of the correct value.
• 1/√3 and 2/√3 involve sqrt(3) and are not related to the 45 degree ratios in this case.
• 1 would be the result of tan 45° alone, not the product with sin 45°.
• Only 1/√2 matches sin 45° × tan 45°.

Common Pitfalls:

• Confusing sin 45° with cos 45°. For 45 degrees they are equal but sometimes students misremember the exact form.
• Thinking tan 45° is something other than 1 and thus changing the product incorrectly.
• Misinterpreting the right angle location and trying to use triangle side lengths instead of known trigonometric values.

Final Answer:
1/√2